Corollary 8.3.4 (Eilenberg–MacLane modules). For every commutative ring \(R\), there is a symmetric monoidal equivalence \[ \D (R)\xrightarrow {\ \simeq \ }\Mod _{HR}(\Sp ). \] Under this equivalence, the functor \(H_R=\hom _{\D (R)}(R[0],-)\) agrees with the forgetful functor \(\Mod _{HR}\to \Sp \), and the lax symmetric monoidal structure on \(H_R\) sends the unit \(R[0]\) to the Eilenberg–MacLane ring spectrum \(HR\) from Corollary 8.3.1.
Proof. Equip \(\D (R)\) with the symmetric monoidal structure of Proposition 20.2.2. Its unit \(R[0]\) is a compact generator by Corollary 20.2.5. Moreover, \[ \pi _n\hom _{\D (R)}(R[0],R[0])\cong H_n(R[0]) \] by Corollary 6.4.5. This is \(R\) for \(n=0\) and zero otherwise. Thus the resulting commutative endomorphism ring spectrum is concentrated in degree \(0\). The adjunction \(\pi _0\colon \Sp _{\geq 0}\rightleftarrows \Ab \noloc H\) is symmetric monoidal by Proposition 8.2.6, Corollary 8.3.1. The induced localization of commutative algebras from Part II (Example 14.5.5), together with the fully faithful symmetric monoidal inclusion \(\Sp _{\geq 0}\hookrightarrow \Sp \), identifies commutative ring spectra concentrated in degree \(0\) with ordinary commutative rings. The multiplication on \(\pi _0\) of the endomorphism ring spectrum is composition of endomorphisms of \(R[0]\), hence the ordinary multiplication of \(R\). It is therefore the Eilenberg–MacLane ring spectrum \(HR\). The result now follows from Theorem 19.5.6. The description of \(H_R\) is built into the comparison functor in that theorem. □
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